To sleep on the sheets is idiomatic, e.g., "Have these sheets been slept on?" is a way of asking whether the sheets are clean.
He sleeps under clean sheets. = Han sover under rene laken. It would be unusual to sleep under more than one sheet.
"Between the sheets" literally means "in bed" but is also is colloquially used to refer to things that happen in bed, so between isn't a great choice either!